# Non-convergent sequence multiplied by a sequence with a zero limit

Let ($t_n$) be a sequence which converges to $0$: $$\lim_{n\to\infty} t_n=0$$ Let ($s_n$) be a sequence which is bounded both from above and below. Show $$\lim_{n\to\infty} t_n s_n=0$$

I'm not totally sure how to approach this problem. Because $s_n$ doesn't necessarily have a limit I've tried approaching it by bringing up the definitions of boundedness but I'm a little lost. What would be the best starting point to go for here?

• The boundedness condition just means $\forall n,\ a\le s_n\le b$ for some constants $a,b$. Also if $c$ is a constant $c\,t_n\to 0$. – zwim Oct 12 '17 at 4:46

Squeeze Theorem. $$\ \ \ \ \ \ \ \ \ \$$
• No, $\{s_n\}$ does not have to be convergent. – Martin Argerami Oct 12 '17 at 5:21